
where \(b^*(\widetilde{c}^t)\mathrel {\mathop {:}}=\operatornamewithlimits{arg\,max}_{b\in [0,1]} \mathsf {Wealth}_t^b(\widetilde{c}^t)\) .
The second term of (REF ) can be bounded by an explicit regret expression (see [1]}), and thus, in principle, solving the maximization problem leads to a confidence sequence.
Since, however, the lower bound is not a quasiconvex function as a function of \(m\) , [1]} came up with an algorithm, dubbed as PRECiSE, that numerically computes the intervals based on the final lower bound (REF ).
While this algorithm gives empirically tighter intervals for small-sample regime and comparable performance in a long run compared to the existing methods of [3]}, the algorithm and analysis are rather complicated by nature.
Moreover, the PRECiSE algorithm still suffers the same per-round time complexity \(O(t)\)  of Cover's UP, since when computing \(b^*(\widetilde{c}^t)\) , maximizing \(\mathsf {Wealth}_t^b(\widetilde{c}^t)\)  necessitates to evaluate the function which has \(t\)  summands.

